Devlog #78 – Wave 58: Galton Board, BZ Reaction & Turing Machine

New Simulations

🎯 Galton Board — Binomial Distribution & Central Limit Theorem

The bean machine

Francis Galton's bean machine (1889) is a physical demonstration of the Central Limit Theorem. Balls drop from a single slot at the top and encounter n rows of pegs. At each peg a ball deflects left with probability p or right with probability q = 1 − p. If we count the number of rightward deflections k, the resulting distribution is the binomial:

P(k) = C(n, k) · p^k · (1-p)^(n-k)     k = 0, 1, ..., n

The mean is μ = n·p and standard deviation σ = √(n·p·q). By the Central Limit Theorem, as n increases this binomial distribution converges to a normal distribution N(np, npq) regardless of the value of p. The simulator shows the binomial PMF as a dashed curve and the smooth normal approximation in teal, both updating live as balls fall.

Simulation design

Balls are animated through the peg grid. Each ball's bin destination is pre-computed at spawn from a Bernoulli sequence, so it follows the exact binomial distribution. The left panel shows the animated cascade; the right panel renders a real-time histogram of bin counts with the theoretical overlay.

• Rows n: 4–16 (up to 17 bins)
• Probability p: 0.1–0.9 (for asymmetric distributions)
• Ball rate: 1–20 balls per animation frame
• Live statistics: observed mean, observed σ, theoretical σ
• Drop 100 balls instantly to fill the histogram quickly

🌀 Belousov-Zhabotinsky Reaction — Excitable-Medium CA

The chemistry behind the spiral

The Belousov-Zhabotinsky reaction is an oscillating chemical system discovered by Boris Belousov in 1951 (and independently by Anatol Zhabotinsky in 1961). A mixture of malonic acid, sodium bromate and a cerium or ferroin catalyst spontaneously cycles between oxidized and reduced states, producing concentric colour waves visible to the naked eye. The Oregonator is the standard kinetic model, but a simpler cellular automaton captures the essential behaviour.

Greenberg-Hastings model

The 3-state excitable-medium CA (Greenberg & Hastings, 1978) uses three cell states:

• Resting (0) — a cell becomes excited if the number of excited nearest-neighbours in the 8-cell Moore neighbourhood meets or exceeds the threshold T.
• Excited (1) — the cell transitions to the refractory state on the next step. This is the "wave front."
• Refractory (2 … k) — the cell counts down k steps back to resting. This creates the "refractory tail" that prevents waves from reversing direction and forces spiral or concentric patterns to persist.

Initial asymmetric seeds produce spiral wave pairs. Spontaneous ignition (a tunable rate) seeds new wave centres over time. All rendering uses ImageData pixel arrays for performance, supporting grids up to 300×300 at 60 fps.

🖥️ Turing Machine — Computability Made Visible

Why simulate a Turing machine?

Alan Turing's 1936 paper "On Computable Numbers" introduced a hypothetical machine that reads and writes symbols on an infinite tape, one cell at a time, using a finite set of rules. Despite its simplicity, the Church-Turing thesis states that any effectively computable function can be computed by such a machine. Visualising execution step by step makes abstract concepts like accepting/rejecting states, halting, and the transition function concrete and graspable.

Five built-in programs

• Binary increment — scans to the end, adds 1 with carry propagation. Input 1011 → 1100.
• Unary addition — replaces the + separator with a 1 and erases the trailing 1. Input 111+11 → 11111.
• Palindrome checker — marks and matches outer characters inward over alphabet {a, b}. Accepts abba, rejects abab.
• String copy — copies a unary string past a separator: 111 → 1110111.
• 3-state Busy Beaver — the champion 3-state BBM writes 6 ones in 14 steps before halting. Start on a blank tape.

Tape display

The canvas renders a sliding window of tape cells centred on the read/write head. The active cell is highlighted with a purple glow; blank cells show a □ symbol. The transition table scrolls to highlight the rule executed at each step. The head pointer turns green on accept, red on reject.

What's Next

Wave 59 will continue expanding coverage across category gaps. High-priority candidates include stochastic resonance (noise-enhanced signal detection in threshold systems), elastic waves (longitudinal and transverse wave propagation with reflection and interference), and cell growth and morphogenesis (reaction-diffusion Turing patterns driving cell differentiation).